On the localization principle for the automorphisms of pseudoellipsoids

We show that Alexander's extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $f$ of a pseudoellipsoid $\E^n_{(p_1, ..., p_{k})} \= \{z \in \C^n : \sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + ... + |z_n|^{2 p_{k}}<1 \}$, provided that $f$ is defined on a region $\U \subset \E^n_{(p)}$ such that: i) $\partial \U \cap \partial \E^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\U \cap \{z_i = 0 \} \neq \emptyset$ for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.

When k = 0, we assume E n (p) to be the unit ball B n = { z ∈ C n : |z| < 1 }. Now, let us consider the following definition.
Definition 1.1. We call local automorphism of E n (p) any biholomorphic map f : U 1 ⊂ E n (p) → U 2 ⊂ E n (p) between two connected open subsets of E n (p) such that: a) each of the intersections ∂U i ∩ ∂E n (p) , i = 1, 2, contains a boundary open set Γ i ⊂ ∂E n (p) ; b) there exists at least one sequence { x k } ⊂ U 1 which converges to a point x o ∈ Γ 1 , which is not a limit point of ∂U 1 ∩ E n (p) , and so that { f (x k ) } converges to a pointx o ∈ Γ 2 , which is not a limit point of ∂U 2 ∩ E n (p) . We say that a local automorphism f : U 1 ⊂ E n (p) → U 2 ⊂ E n (p) extends to a global automorphism of E n (p) if there exists some F ∈ Aut(E n (p) ) such that F | U1∩E n By a celebrated theorem of Alexander and its generalization obtained by Rudin ( [Al, Ru]), when E n (p) = B n , any local automorphism extends to a global one. This crucial extendibility result is often quoted as localization principle for the automorphisms of B n and it has been extended or established under different but similar hypotheses, for a wide class of domains besides the unit balls (see e. g. [DS,Pi,Pi1]). On the other hand, even if it is known that the pseudoellipsoids E n share many useful properties with B n for what concerns the global automorphisms and the proper holomorphic maps (see f.i. [We, La, LS, DS]), some simple examples show that Alexander's theorem cannot be true in full generality for a pseudoellipsoid E n (p) different from B n (see e.g. Example 3.4 below). Nonetheless, for each E n (p) , it is possible to determine, precisely and in an efficient way, the class of local automorphisms that can be extended to global ones. In this short note we give a characterization of such local automorphisms by means of the following generalization of Alexander's theorem.
be a local automorphism of a pseudoellipsoid E n (p) , with p = (p 1 , . . . , p k ), and satisfying the following two conditions: i) there exists a sequence {x i } as in (b) of Definition 1.1, whose limit point Then f extends to a global automorphism f ∈ Aut(E n (p) ).
We point out that the set ∂E n (p) ∩ n i=n−k+1 { z i = 0 } coincides with the set of points of Levi degeneracy of ∂E n (p) . So, Theorem 1.2 can be roughly stated saying that f is globally extendible as soon as it admits an holomorphic extension to some open subset U ⊂ E n (p) , which intersects each of the hyperplanes containing the Levi degeneracy set of ∂E n (p) and, at the same time, the boundary ∂U contains an open set of strongly pseudoconvex points of ∂E n (p) . From next Example 3.4, it will be clear that such hypotheses can be considered as optimal.
The properties of the pseudoellipsoid used in the proof are basically just two: (1) It admits a finite ramified covering over the unit ball; (2) Its automorphisms are "lifts" of the automorphisms of the unit ball that preserve the singular values of the covering. Since (2) is a consequence of (1), it is reasonable to expect that a similar result should be true for any arbitrary ramified covering of the unit ball.
About this more general problem, we refer to [KLS, KS] for what concerns the classification of the domains in C 2 that admit a ramified holomorphic covering over B 2 .

On the automorphisms of the unit ball
First of all, we need to recall some basic facts on the automorphisms of the unit ball. Let us denote byî : C n → CP n the canonical embeddinĝ where we denote by <, > the pseudo-Hermitian inner product on C n+1 defined by It is also known that a holomorphic map F : B n → B n is an automorphism of B n if and only if the corresponding mapF =î • F •î −1 :B n →B n is a projective linear transformation which preserves the quadric ∂B n = { [w] : < w, w >= 0 } (see e.g. [Ve]). This means thatF is of the form where A is a matrix in SU n,1 , i.e. such that A t I n,1 A = I n,1 and with det A = 1.
The correspondence F →F =î • F •î −1 gives an isomorphism between Aut(B n ) The identification of the elements of Aut(B n ) with the corresponding projective linear transformations is often quite useful, for instance in order to establish the following fact (see also [We], §6). This implies that the matrix A which determines the projective transformationF is of the form is an automorphisms ofB n−k ⊂ CP n−k \ { w n−k+1 = 0 }.

Proof of Theorem 1.2
First of all, we need to introduce the following notation. For any p = (p 1 , . . . , p k ), we will use the symbol π (p) to denote the map π (p) : C n → C n , π (p) (z) = (z 1 , . . . , z n−k , z p1 n−k+1 , . . . , z p k n ) . We recall that the restriction π (p) gives a proper holomorphic map π (p) : Secondly, we need to recall a useful theorem by Forstneric and Rosay ([FR]). Given a domain D ⊂ C n , we say that a boundary point z o ∈ ∂D satisfies the condition (P ) if: -∂D is of class C 1+ε near z o for some ε > 0; -there exist a continuous negative plurisubharmonic function ρ on D and a neighborhood U of z o so that ρ(z) ≥ −c d(z, ∂D) at all points of U ∩ D for some constant c > 0. Theorem 1.1 and some related remarks of [FR] can be summarized as follows.
Theorem 3.1. Let h : D → D ′ be a proper holomorphic map between two domains of C n and let z o ∈ ∂D be a point that satisfies the condition (P).
If there exists a sequence {z j } ⊂ D so that lim j→∞ z j = z o and lim j→∞ h(z j ) = z o for someẑ o ∈ ∂D ′ at which ∂D ′ is C 2 and strictly pseudoconvex, then h extends continuously to all points of neighborhood V of z o in D.
We may now prove the following lemma.
Lemma 3.2. Let f : U 1 ⊂ E n (p) → U 2 ⊂ E n (p) be a local automorphism of a pseudoellipsoid E n (p) with p = (p 1 , . . . , p k ) and assume that i) there exists a sequence {x i } as in (b) of Definition 1.1, whose limit point x o ∈ ∂E (p) is Levi non-degenerate; ii) for any n − k + 1 ≤ i ≤ n, the intersection U 1 ∩ { z i = 0 } is not empty. Then, up to composition with a coordinate permutation (z 1 , . . . , z n ) → (z σ(1) , . . . , z σ(n) ) , (3.1) the map f sends the points of the hyperplane { z i = 0 } into the same hyperplane for any n − k + 1 ≤ i ≤ n.
Proof. In all the following we will use the symbols Γ i , x o andx o with the same meaning as in Definition 1.1. First of all, notice thatx o ∈ Γ 2 ⊂ ∂U 2 satisfies the condition (P) and hence, by Theorem 3.1, for any sufficiently small ball B ε (x o ), centered atx o and of radius ε, the holomorphic map f −1 : U 2 → U 1 extends continuously to all points of B ε (x o ) ∩ Γ 2 . In particular, we may assume that Pick a Levi non-degenerate pointx ′ o ∈ B ε (x o ) ∩ Γ 2 and consider a sequence and by Theorem 3.1 applied to f and f −1 , there is no loss of generality if we assume that x o andx o are both Levi non-degenerate and that, for any sufficiently small ε 1 > 0, the map f extends continuously to a map which is an homeomorphism onto its image.
Since the complex Jacobian matrices Jπ (p) xo and Jπ (p) xo are of maximal rank (recall that x o andx o ∈ ∂E n (p) are both Levi non-degenerate), from the fact that x o is not a limit point of ∂U 1 ∩ E n (p) and by the continuity of f and f −1 around x o and x o , respectively, we may choose ε 1 and ε 2 so that: a) π (p) | Bε 1 (xo) and π (p) | Bε 2 (xo) are both biholomorphisms onto their images; are local automorphisms of E n (p) and of the unit ball, respectively. By Rudin's generalization of Alexander's theorem ( [Ru]), this implies thatf extends to a global automorphism of B n , which we denote byf as well. By construction, for any z ∈ U ′ 1 = π (p)−1 (V 1 ), we havẽ f • π (p) (z) = π (p) • f (z) , (3.2) but since both sides have an holomorphic extension on U 1 , we get that (3.2) must be true also for any z in such larger set. In particular, Since for any z ∈ U 1 , det J(f )| z = 0 and equality (3.3) implies that, for any n−k+1 ≤ i ≤ n and z ∈ U 1 ∩{ z i = 0 }, the value of J(π (p) )| f (z) is 0. By (3.4), this means that f (U 1 ∩ { z i = 0 }) is contained in the union n j=n−k+1 { z j = 0 }. Indeed, it is contained in exactly one of the hyperplanes {z j = 0}, because f is a biholomorphism and consequently f (U 1 ∩ { z i = 0 }) is an irreducible analytic variety. From this the conclusion follows.
We proceed by defining a rule that associates an automorphism of B n with any local automorphism of a pseudoellipsoid (see also [We], §6). Given a local automorphism f : U → C n of E n (p) , pick a point x o ∈ U ∩ ∂E n (p) for which (b) of Definition 1.1 holds and determine a small ball B ε (x o ) centered in x o as in the proof of the previous lemma. Then, we denote byf ∈ Aut(B n ) the global automorphism of the unit ball that extendsf . By the identity principle of the holomorphic maps, such automorphismf depends only on f and it will be called the (global) automorphism of B n associated with f .
With the help of such correspondence, we may state the following criterion for extendibility of local automorphisms.
Example 3.4. Letf ∈ Aut(B n ) be an automorphism which does not satisfies (2.3) for some n − k + 1 ≤ j ≤ n. Pick a point w o ∈ ∂B ∩ { n j=n−k+1 z j = 0 } so that also its imagef (w o ) is in ∂B ∩ { n j=n−k+1 z j = 0 }. Then, let z o ∈ ∂E n (p) so that π (p) (z o ) = w o and consider a connected neighborhood U of z o with the following two properties: a) π (p) | U is a biholomorphism between U and its image π (p) (U); b)f (π (p) (U)) does not intersect n j=n−k+1 z j = 0 (a sufficiently small neighborhood U surely satisfies both requirements). Then, we may consider the map By construction, f is a local automorphism of E n (p) and its associated automorphism of Aut(B n ) isf . By the hypotheses onf and by Proposition 3.3, f cannot extend to a global automorphism of E n (p) .